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Question
If \[\vec{A} = 2 \vec{i} + 3 \vec{j} + 4 \vec{k} \text { and } \vec{B} = 4 \vec{i} + 3 \vec{j} + 2 \vec{k}\] find \[\vec{A} \times \vec{B}\].
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Solution
Given:
\[\vec{A} = 2 \hat {i} + 3 \hat {j} + 4 \hat {k}\] and
\[\vec{B} = 4 \hat {i} + 3 \hat {j} + 2 \hat {k} \]
The vector product of \[\vec{A} \times \vec{B}\]
\[\vec{A} \times \vec{B} = \begin{vmatrix}\hat {i} & \hat {j} & \hat {k} \\ 2 & 3 & 4 \\ 4 & 3 & 2\end{vmatrix}\]
\[ = \hat {i} \left( 6 - 12 \right) - \hat {j} \left( 4 - 16 \right) + \hat {k} \left( 6 - 12 \right)\]
\[ = - 6 \hat {i} + 12 \hat {j} - 6 \hat {k}\]
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